The pure spin currents, i.e., the counterflow of particles with opposite spin orientations, can be optically injected in semiconductors. Here, we develop a phenomenological theory, which describes the polarization dependencies of spin currents excited by linearly polarized light in bulk semiconductors and quantum well structures of various symmetries. We present microscopic descriptions of the pure spin photocurrents for interband optical transitions in undoped quantum wells as well as for direct intersubband and indirect intrasubband (Drude-like) transitions in n-doped quantum well structures. We also demonstrate that pure spin currents can be generated in structures of sufficiently low symmetries by simple electron gas heating. The theoretical results are compared with recent experimental observations.